charge-transfer excitations made ``simple'' by subsystem dftcharge-transfer excitations made...

Charge-Transfer Excitations Made “Simple” bySubsystem DFT

Michele Pavanello & Pablo Ramos

Department of ChemistryRutgers University-Newark

Newark, NJ

ADF Webinar Series, Feb. 24th, 2015

Michele Pavanello & Pablo Ramos Subsystem DFT

What is PRG interested in?

Ab-Initio Modeling of Realistically Sized Systemswith subsystem DFT!

Charge transfer phenomenaElectronic structure and couplingsApplication to biosystems, and organic electrode materials→ DNA, Sulfite Oxidase (SCM booth)→ Organic electrode materials based on croconate crystals

Electronic structure of periodic systems [Davide Ceresoli]AIMD, RT-TDDFT, Ehrenfest→ RT-TDDFT, Ehrenfest (Thursday afternoon)→ Ground state (Poster)Application to molecules at surfaces, liquids, ionic crystals, etc

Linear-response TD-DFT and vdW [Henk Eshuis]

What is PRG interested in?

Ab-Initio Modeling of Realistically Sized Systemswith subsystem DFT!

Charge transfer phenomenaElectronic structure and couplingsApplication to biosystems, and organic electrode materials→ DNA, Sulfite Oxidase (SCM booth)→ Organic electrode materials based on croconate crystals

Electronic structure of periodic systems [Davide Ceresoli]AIMD, RT-TDDFT, Ehrenfest→ RT-TDDFT, Ehrenfest (Thursday afternoon)→ Ground state (Poster)Application to molecules at surfaces, liquids, ionic crystals, etc

Linear-response TD-DFT and vdW [Henk Eshuis]

How to Calculate Electronic Couplings?

Adiabatic-to-Diabatic transformationAll-electron methodM. Newton, R. Cave, A. Voityuk, C. Hsu, J. Subotnik, . . .Usually associated with wavefunction methodsCan be used in conjunction with TD-DFT, but self-interaction errorcomplicates thingsAs accurate as the method for computing the adiabatic states

Fragment OrbitalsFrozen core methodF. Grozema, M. Ratner, J. L. Bredas, J. Blumberger, M. Elstner, . . .The most common methodAbout 20%-40% accurate [J. Chem. Phys. 140 , 104105 (2014)]

Frozen Density Embedding (FDE-ET)All-electron methodM. Pavanello, J. Neugebauer, L. VisscherNew method [J. Chem. Phys. 138 , 054101 (2013)]Better than 10% accurate [Submitted to J. Phys. Chem. B]

Tackling large systems with subsystem DFT: FDE

Split the system into (smaller) interacting subsystemsPartition of the total electron density into subsystem contributions1

ρ(r) = ρI(r)+ ρII(r) ρI/II(r) =occI/II∑

i

|φI/II,i(r)|2

The energy functional is almost additive: E[ρ] ' E[ρI ] + E[ρII ]

E[ρ] = E[ρI ] + E[ρII ]+Tnadds [ρI , ρII ] + Enaddxc [ρI , ρII ] + V

naddCoul [ρI , ρII ]

Fnadd[ρI , ρII ] = F[ρ]− F[ρI ]− F[ρII ]

Frozen Density Embedding (FDE): coupled Kohn–Sham equations foreach subsystem

δE[ρI + ρII ]δ ρI

→[−1

2∇2 + vIKS(r) + vIemb(r)

]φIi (r) = ε

Iiφ

Ii (r)

1P. Cortona, PRB 44, 8454 (1991), Wesolowski & Warshel, JPC 97, 8050, (1993)Michele Pavanello & Pablo Ramos Subsystem DFT

Split the system into (smaller) interacting subsystems

Partition of the total electron density into subsystem contributions1

i

|φI/II,i(r)|2

→[−1

]φIi (r) = ε

Iiφ

Ii (r)

i

|φI/II,i(r)|2

→[−1

]φIi (r) = ε

Iiφ

Ii (r)

i

|φI/II,i(r)|2

→[−1

]φIi (r) = ε

Iiφ

Ii (r)

i

|φI/II,i(r)|2

→[−1

]φIi (r) = ε

Iiφ

Ii (r)

i

|φI/II,i(r)|2

→[−1

]φIi (r) = ε

Iiφ

Ii (r)

i

|φI/II,i(r)|2

→[−1

]φIi (r) = ε

Iiφ

Ii (r)

KS equations in Frozen Density Embedding (FDE)a

vIKS(r) = vInuc(r) +

∫ρI(r′)|r− r′|

dr′ +δExc[ρI ]δ ρI

vIemb(r) = vIInuc(r) +

∫ρII(r′)|r− r′|

dr′ +δEnaddxc [ρI , ρII ]δ ρI(r′)

+δTnadd[ρI , ρII ]δ ρI(r′)

.

Non-additive functionals approximants

Enaddxc [ρI , ρII ]: GGA functionalsTnadds [ρI , ρII ]: orbital-free GGA expressions

. . . if use LDA:

δTnadd[ρI , ρII ]δρI(r′)

=53

CF{[ρI (r′) + ρII (r′)

]2/3 − [ ρI (r′) ]2/3}

∫ρI(r′)|r− r′|

.

. . . if use LDA:

=53

]2/3 − [ ρI (r′) ]2/3}

∫ρI(r′)|r− r′|

.

Enaddxc [ρI , ρII ]: GGA functionals

Tnadds [ρI , ρII ]: orbital-free GGA expressions

. . . if use LDA:

=53

]2/3 − [ ρI (r′) ]2/3}

∫ρI(r′)|r− r′|

.

. . . if use LDA:

=53

]2/3 − [ ρI (r′) ]2/3}

∫ρI(r′)|r− r′|

.

. . . if use LDA:

=53

]2/3 − [ ρI (r′) ]2/3}

Donor–Acceptor PictureDiabatic states

Model Hamiltonian

Ĥ = H′ii|i〉〈i|+ H′if |i〉〈f |+ H′ff |f 〉〈f |

Basis set for a hole transfer reaction

Initial state →[

D]+− (bridge)−

[A]

Final state →[

D]− (bridge)−

[A]+

The coupling

H′if =〈D+A|Ĥ|DA+

〉

=

〈[D]+− (bridge)−

[A]∣∣∣∣Ĥ∣∣∣∣[D]− (bridge)− [A]+

〉

Model Hamiltonian

Initial state →[

D]+− (bridge)−

[A]

Final state →[

D]− (bridge)−

[A]+

The coupling

〉

=

[A]∣∣∣∣Ĥ∣∣∣∣[D]− (bridge)− [A]+

〉

Model Hamiltonian

Initial state →[

D]+− (bridge)−

[A]

Final state →[

D]− (bridge)−

[A]+

The coupling

〉

=

[A]∣∣∣∣Ĥ∣∣∣∣[D]− (bridge)− [A]+

〉

Model Hamiltonian

Initial state →[

D]+− (bridge)−

[A]

Final state →[

D]− (bridge)−

[A]+

The coupling

〉

=

[A]∣∣∣∣Ĥ∣∣∣∣[D]− (bridge)− [A]+

〉

Model Hamiltonian

Initial state →[

D]+− (bridge)−

[A]

Final state →[

D]− (bridge)−

[A]+

The coupling

〉

=

[A]∣∣∣∣Ĥ∣∣∣∣[D]− (bridge)− [A]+

〉

Model Hamiltonian

Initial state →[

D]+− (bridge)−

[A]

Final state →[

D]− (bridge)−

[A]+

The coupling

〉=

[A]∣∣∣∣Ĥ∣∣∣∣[D]− (bridge)− [A]+

〉

Diabatic representation: making charge-localizedstates with FDE

Subsystem 2Subsystem 1

+0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣Ĥ

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣Subsystem 2

Subsystem 1

0+

Electronic Couplings With Subsystem DFTa: FDE-ET

Hartree–Fock

Hif = 〈Φi|Ĥ|Φf 〉 = E[ρ(if )(r)

]Sif

where Sif = 〈Φi|Φf 〉, and

ρ(if )(r) =∑

kl

φ(i)k (r)

(S(if )

)−1klφ(f )l (r)

where S(if )kl = 〈φ(i)k |φ

(f )l 〉

Subsystem DFT formalism

Hif = E[ρ(if )

]Sif . . . an approximation

Hif =

[( NS∑I

E[ρ(if )I (r)

])+ Tnadds

[{ρ(if )I

}I=1,...,NS

]+ Enaddxc

[{ρ(if )I

}I=1,...,NS

]]Sif

It’s linear scaling!!

a MP, Van Voorhis, Visscher, Neugebauer, JCP 138, 054101 (2013)

Does FDE-ET work?

Hole transfer in the ethylene dimer radical cation

Excitation Energy (eV)R Sif This Worka CCSD(T)b CCSD(T)c FS-CCSDd

3.0 1.60E-1 1.833 1.913 1.939 1.9474.0 6.27E-2 0.521 0.527 0.558 0.5556.0 6.55E-3 0.037 0.024 0.038 0.0359.0 1.82E-4 6E-4 0.009 0.004 3E-410.0 5.31E-5 1E-4 0.009 0.003 5E-515.0 8.75E-8 2E-6 0.009 0.004 1E-7

a PW91/TZPb EOM-CCSD(T)/6-311G(d,p) ROHFc EOM-CCSD(T)/aug-cc-pVDZ ROHFd FS-CCSD(T)/aug-cc-pVDZ UHF

MP, Van Voorhis, Visscher, Neugebauer, JCP 138, 054101 (2013)

The Superexchange Hole Transfer in DNA

G

T T T

G

Ener

gy

hole transfer

a Ramos and MP, JCTC 10, 2546 (2014)

Calculation of hole transfer couplings with FDE-ET

DNA base pairs: distance dependence of excitation energies andcouplingsa

3 4 5 6 7 8Separation (Ang)

0

0.5

1

1.5

2

Ex

cita

tio

n E

ner

gy

(eV

)Guanine dimerGuanine Thymine

Thymine dimer

2.5 < β < 3.5

a Ramos and MP, JCTC 10, 2546 (2014)

Donor-Bridge-Acceptor Effective Couplingsa withFDE-ETb

Hole transfer in DNA oligomers: GC

(TA

)n

GC

D

A

En

erg

y

Single strand

Double strand

Deep Tunnelling

VDA ' HDA − SDAHDD + HAA

2

decays too fast: β ' 3

a A. Nitzan, Chemical Dynamics in Condensed Phases, Oxford University Press (2006)b Ramos and MP, JCTC 10, 2546 (2014)

Hole transfer in DNA oligomers: GC

(TA

)n

GC

BB

B

D

A

En

erg

y

Single strand

Double strand

Deep Tunnelling

VDA ' HDA − SDAHDD + HAA

2

decays too fast: β ' 3

a A. Nitzan, Chemical Dynamics in Condensed Phases, Oxford University Press (2006)b Ramos and MP, JCTC 10, 2546 (2014)

5 10 15Distance / Ang

-25

-20

-15

-10

-5

2ln

|V|

double strand

single strand

Superexchange mechanism

V ′DA(E) = VDA + VDBGB(E)VBA

accounts for a lower tunnellingbarrier than the vacuum: β ' 1

much closer to experimentc

a A. Nitzan, Chemical Dynamics in Condensed Phases, Oxford University Press (2006)b Ramos and MP, JCTC 10, 2546 (2014)c B. Giese, Annu. Rev. Biochem. 71, 51-70 (2002)

Effective Couplings Between Charge SeparatedStatesa with Subsystem DFT

Charge separation (ground state)

D + A→ D+ + A−

Charge separation (excited state)

D∗ + A→ D+ + A−

HOMOs for the ethylene dimer in the charge separated state

The FDE calculation needs extra care.

The orbitals of the anions might not becorrect due to self-interaction. If a self-interaction corrected functional is used,these issues are drastically reduced.

a Solovyeva, Pavanello, Neugebauer, JCP, 140, 164103 (2014)

D + A→ D+ + A−

D∗ + A→ D+ + A−

a Solovyeva, Pavanello, Neugebauer, JCP, 140, 164103 (2014)

D + A→ D+ + A−

D∗ + A→ D+ + A−

a Solovyeva, Pavanello, Neugebauer, JCP, 140, 164103 (2014)Michele Pavanello & Pablo Ramos Subsystem DFT

Practical Calculations: outline

Initial FDE calculations

Compute D+ + Adiabat ACompute D + A+

diabat BThe input file must beprepared manuallyPyADF support is beingdevelopped (L. Vissher)

In output:1 T21 file for the isolated D:t21.iso.rho11 T21 file for the isolated A:t21.iso.rho12 T21 files for the embedded D:fragA1.t21, fragB1.t212 T21 files for the embedded A:fragA2.t21, fragB2.t21

FDE-ET calculationOnce the FDE calculations for the A and B diabats are complete, theFDE-ET calculation can follow. The keyword needed is

ElectronTransfer

Practical Calculations: Making the FDE-ET input file

Fragment key need this setup to loadsome predefined parameters from theisolated calculation (basis sets, AOmaps, . . . )

FRAGMENTSrho1 t21.iso.rho1rho2 t21.iso.rho2

END

ElectronTransfer key. NumFrag speci-fies the number of fragments involved.InvThr is a threshold for handlingquasi-orthogonality. Joint/Disjoint: thedefault is Joint. Disjoint is not recom-mended yet.

ELECTRONTRANSFERNumFrag 2{Joint|Disjoint}{InvThr 1.0e-3}

END

See example file at:

$ADFHOME/examples/adf/ElectronTransfer_FDE_H2O

Practical Calculations: Making the FDE-ET input file

Fragment key need this setup to loadsome predefined parameters from theisolated calculation (basis sets, AOmaps, . . . )

FRAGMENTSrho1 t21.iso.rho1rho2 t21.iso.rho2

END

ElectronTransfer key. NumFrag speci-fies the number of fragments involved.InvThr is a threshold for handlingquasi-orthogonality. Joint/Disjoint: thedefault is Joint. Disjoint is not recom-mended yet.

ELECTRONTRANSFERNumFrag 2{Joint|Disjoint}{InvThr 1.0e-3}

END

See example file at:

Practical Calculations: The output file

The output file from:

Is the following:-------------------------------------------------------------------==> WARNING: one less electron in the transition density matrix applying the Penrose inversion of the N-1 dimensional subspace

The output file from:

Is the following:-------------------------------------------------------------------==> WARNING: one less electron in the transition density matrix applying the Penrose inversion of the N-1 dimensional subspace

Let’s look at another output file:

============ Electron Transfer RESULTS ===================Electronic Coupling = 0.344469 eVElectronic Coupling = 2778.314671 cm-1H11-H22 = 0.000574 eVExcitation Energy = 0.688937 eVOverlap = 0.066531H11 H22 H12 = -1384.925285303533 -1384.925306415906 -1385.114726153074 EhS11 S22 S12 = 0.880467985626 0.880488669229 -0.058578952462=========== END Electron Transfer RESULTS ================

Things to note

H11-H22 is the energydifference of the diabaticstatesExcitation Energy maydiffer from H11-H22

Overlap is the normalizedoverlap

S′ij =Sij√

Sii × Sjj

Let’s look at another output file:============ Electron Transfer RESULTS ===================Electronic Coupling = 0.344469 eVElectronic Coupling = 2778.314671 cm-1H11-H22 = 0.000574 eVExcitation Energy = 0.688937 eVOverlap = 0.066531H11 H22 H12 = -1384.925285303533 -1384.925306415906 -1385.114726153074 EhS11 S22 S12 = 0.880467985626 0.880488669229 -0.058578952462=========== END Electron Transfer RESULTS ================

Things to note

S′ij =Sij√

Sii × Sjj

Let’s look at another output file:============ Electron Transfer RESULTS ===================Electronic Coupling = 0.344469 eVElectronic Coupling = 2778.314671 cm-1H11-H22 = 0.000574 eVExcitation Energy = 0.688937 eVOverlap = 0.066531H11 H22 H12 = -1384.925285303533 -1384.925306415906 -1385.114726153074 EhS11 S22 S12 = 0.880467985626 0.880488669229 -0.058578952462=========== END Electron Transfer RESULTS ================

Things to note

S′ij =Sij√

Sii × Sjj

FDE-ET in summary

What has been done with FDE-ETApplied to hole transfer and charge separation in pilot calculations.Tested for hole transfer.

RecommendationsUse the PW91k nonadditive Kinetic Energy functional in the FDEcalculations.No particular problems applying FDE-ET to cations (hole transfer)→ InvThr: might need 1.0e-2 if a near orthogonality is suspected.Application to anions is not recommended, however it is possible. UseSIC functionals in the FDE part, and then a GGA functional in theElectronTransfer part.Application to charge separation (+/-) and to other spin states (e.g.triplet) is not available in the released code. Perhaps in the next ADFrelease.

FDE-ET in summary

Pavanello Research Group

I am looking for. . .

PhD & Master’s students

Available Projects

Non-adiabatic dynamics with Subsystem DFTVan der Waals interactions between subsystemsSubsystem DFT applied to Materials research. . .

Rutgers University, Newark, NJ

20’ from NYCFun learning environment

Acknowledgments

Postdocs, Students & Collaborators

Dr. Debalina SinhaPablo RamosMarc Mankarious

Prof. J. Neugebauer(WWU, Münster)Prof. L. Visscher (VU,Amsterdam)Prof. T. Van Voorhis (MIT)

Funding

Start-up funds fromRutgers-NewarkNSF-CBET-1438493NSF-CNIC-1404739

AMSTERDAM DENSITYFUNCTIONALPetroleum Research Fund(54063-DNI6)ELF fund – State of NewJersey

charge-transfer excitations made ``simple'' by subsystem dftcharge-transfer excitations made...

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